Symbolic Computation of Lax Pairs of Partial Difference Equations using Consistency Around the Cube

• Published in: Foundations of computational mathematics Vol. 13; no. 4; pp. 517 - 544
• Main Authors: Bridgman, T., Hereman, W., Quispel, G. R. W., van der Kamp, P. H.
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.08.2013; Springer; Springer Nature B.V
• Subjects: Applications of Mathematics; Computation; Computational mathematics; Computer Science; Cube; Cubes; Difference equations; Differential equations, Partial; Economics; Lattice theory; Lattices; Linear and Multilinear Algebras; Math Applications in Computer Science; Mathematical analysis; Mathematics; Mathematics and Statistics; Matrix Theory; Numerical Analysis; Partial differential equations; Quadrilaterals; Scalars; Schroedinger equation; Australia; Consistency around the cube; 82B20; 39A10; Lax pairs; 37K10; Integrable lattice equations
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-012-9133-9

A three-step method due to Nijhoff and Bobenko & Suris to derive a Lax pair for scalar partial difference equations (PΔEs) is reviewed. The method assumes that the PΔEs are defined on a quadrilateral, and consistent around the cube. Next, the method is extended to systems of PΔEs where one has to carefully account for equations defined on edges of the quadrilateral. Lax pairs are presented for scalar integrable PΔEs classified by Adler, Bobenko, and Suris and systems of PΔEs including the integrable two-component potential Korteweg–de Vries lattice system, as well as nonlinear Schrödinger and Boussinesq-type lattice systems. Previously unknown Lax pairs are presented for PΔEs recently derived by Hietarinta (J. Phys. A, Math. Theor. 44:165204, 2011 ). The method is algorithmic and is being implemented in Mathematica .